Journal of Combinatorial Designs最新文献

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A generalization of group divisible t $t$ -designs 群可整除t$t$-设计的一个推广
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-08-10 DOI: 10.1002/jcd.21912
Sijia Liu, Yue Han, Lijun Ma, Lidong Wang, Zihong Tian
{"title":"A generalization of group divisible \u0000 \u0000 \u0000 t\u0000 \u0000 $t$\u0000 -designs","authors":"Sijia Liu,&nbsp;Yue Han,&nbsp;Lijun Ma,&nbsp;Lidong Wang,&nbsp;Zihong Tian","doi":"10.1002/jcd.21912","DOIUrl":"https://doi.org/10.1002/jcd.21912","url":null,"abstract":"<p>Cameron defined the concept of generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, which generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs and group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, and then point out close connections with various well-known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs and almost completely determine their existence for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation> $t=2,3$</annotation>\u0000 </semantics></math> and small block sizes.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"575-603"},"PeriodicalIF":0.7,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50137077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cameron–Liebler sets for maximal totally isotropic flats in classical affine spaces 经典仿射空间中最大全各向同性平面的Cameron–Liebler集
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-07-19 DOI: 10.1002/jcd.21909
Jun Guo, Lingyu Wan
{"title":"Cameron–Liebler sets for maximal totally isotropic flats in classical affine spaces","authors":"Jun Guo,&nbsp;Lingyu Wan","doi":"10.1002/jcd.21909","DOIUrl":"https://doi.org/10.1002/jcd.21909","url":null,"abstract":"&lt;p&gt;Let &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $ACG(2nu ,{{mathbb{F}}}_{q})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $2nu $&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional classical affine space with parameter &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $e$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; over a &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $q$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-element finite field &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{mathbb{F}}}_{q}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{mathscr{O}}}_{nu }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the set of all maximal totally isotropic flats in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $ACG(2nu ,{{mathbb{F}}}_{q})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, we discuss Cameron–Liebler sets in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;O&lt;/mi&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${{mathscr{O}}}_{nu }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"547-574"},"PeriodicalIF":0.7,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50137650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Exceptional designs in some extended quadratic residue codes 一些扩展二次残差码的例外设计
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-07-18 DOI: 10.1002/jcd.21907
Reina Ishikawa
{"title":"Exceptional designs in some extended quadratic residue codes","authors":"Reina Ishikawa","doi":"10.1002/jcd.21907","DOIUrl":"https://doi.org/10.1002/jcd.21907","url":null,"abstract":"<p>In the present paper, we give proofs of the existence of a 3-design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"496-510"},"PeriodicalIF":0.7,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50152059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
On the existence of k $k$ -cycle semiframes for even k $k$ 关于偶数k$k的k$k$-循环半帧的存在性$
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-07-13 DOI: 10.1002/jcd.21908
Li Wang, Haibo Ji, Haitao Cao
{"title":"On the existence of \u0000 \u0000 \u0000 k\u0000 \u0000 $k$\u0000 -cycle semiframes for even \u0000 \u0000 \u0000 k\u0000 \u0000 $k$","authors":"Li Wang,&nbsp;Haibo Ji,&nbsp;Haitao Cao","doi":"10.1002/jcd.21908","DOIUrl":"https://doi.org/10.1002/jcd.21908","url":null,"abstract":"&lt;p&gt;A &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${C}_{k}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-semiframe of type &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${g}^{u}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${C}_{k}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-group divisible design of type &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;ℬ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${g}^{u}({mathscr{X}},{mathscr{G}},{rm{ {mathcal B} }})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; in which &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${mathscr{X}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the vertex set, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${mathscr{G}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the group set, and the set &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℬ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{ {mathcal B} }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-cycles can be written as a disjoint union &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℬ&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 &lt;mo&gt;∪&lt;/mo&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{ {mathcal B} }}={mathscr{P}}cup {mathscr{Q}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;P&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${mathscr{P}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is partitioned into parallel classes","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"511-530"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50131264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Totally symmetric quasigroups of order 16 16阶全对称拟群
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-07-13 DOI: 10.1002/jcd.21910
Hy Ginsberg
{"title":"Totally symmetric quasigroups of order 16","authors":"Hy Ginsberg","doi":"10.1002/jcd.21910","DOIUrl":"https://doi.org/10.1002/jcd.21910","url":null,"abstract":"<p>We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes of such objects. Totally symmetric quasigroups of orders up to and including 16 that are (respectively) medial, idempotent, and unipotent are also enumerated.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"531-542"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50131265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Enumerating Steiner triple systems Steiner三重系统的枚举
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-07-13 DOI: 10.1002/jcd.21906
Daniel Heinlein, Patric R. J. Östergård
{"title":"Enumerating Steiner triple systems","authors":"Daniel Heinlein,&nbsp;Patric R. J. Östergård","doi":"10.1002/jcd.21906","DOIUrl":"https://doi.org/10.1002/jcd.21906","url":null,"abstract":"<p>Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14,796,207,517,873,771.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 10","pages":"479-495"},"PeriodicalIF":0.7,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21906","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50140412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cycles of quadratic Latin squares and antiperfect 1-factorisations 二次拉丁平方的环与反完美1-因子分解
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-07-10 DOI: 10.1002/jcd.21905
Jack Allsop
{"title":"Cycles of quadratic Latin squares and antiperfect 1-factorisations","authors":"Jack Allsop","doi":"10.1002/jcd.21905","DOIUrl":"https://doi.org/10.1002/jcd.21905","url":null,"abstract":"<p>A Latin square of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> is an <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>×</mo>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $ntimes n$</annotation>\u0000 </semantics></math> matrix of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math> let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>F</mi>\u0000 \u0000 <mi>q</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> ${{mathbb{F}}}_{q}$</annotation>\u0000 </semantics></math> denote the finite field of order <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 <annotation> $q$</annotation>\u0000 </semantics></math>. A quadratic Latin square is a Latin square <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>b</mi>\u0000 </mrow>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal L} }}[a,b]$</annotation>\u0000 </semantics></math> defined by\u0000\u0000 </p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"447-475"},"PeriodicalIF":0.7,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21905","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50127978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The chromatic index of finite projective spaces 有限射影空间的色指数
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-06-30 DOI: 10.1002/jcd.21904
Lei Xu, Tao Feng
{"title":"The chromatic index of finite projective spaces","authors":"Lei Xu,&nbsp;Tao Feng","doi":"10.1002/jcd.21904","DOIUrl":"https://doi.org/10.1002/jcd.21904","url":null,"abstract":"&lt;p&gt;A line coloring of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;PG&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $text{PG}(n,q)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional projective space over GF&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $(q)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, is an assignment of colors to all lines of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;PG&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $text{PG}(n,q)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; so that any two lines with the same color do not intersect. The chromatic index of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;PG&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 \u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 \u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $text{PG}(n,q)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, denoted by &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;′&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mtext&gt;PG&lt;/mtext&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 \u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 \u0000 &lt;mo&gt;,&lt;/","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"432-446"},"PeriodicalIF":0.7,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of small ordered orthogonal arrays 小有序正交阵列的存在性
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-06-19 DOI: 10.1002/jcd.21903
Kai-Uwe Schmidt, Charlene Weiß
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引用次数: 0
Linear and circular single-change covering designs revisited 重新审视线性和圆形单次变更覆盖设计
IF 0.7 4区 数学
Journal of Combinatorial Designs Pub Date : 2023-06-01 DOI: 10.1002/jcd.21885
Amanda Chafee, Brett Stevens
{"title":"Linear and circular single-change covering designs revisited","authors":"Amanda Chafee,&nbsp;Brett Stevens","doi":"10.1002/jcd.21885","DOIUrl":"https://doi.org/10.1002/jcd.21885","url":null,"abstract":"&lt;p&gt;A &lt;i&gt;single-change covering design&lt;/i&gt; (SCCD) is a &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-set &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and an ordered list &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℒ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; ${rm{ {mathcal L} }}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $b$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; blocks of size &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where every pair from &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v,k$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, we use a new recursive construction to solve the existence of circular SCCD(&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v,4,b$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) for all &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and three residue classes of circular SCCD(&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;v&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt; $v,5,b$&lt;/annotation&gt;\u0000 &lt;/semantics","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"405-421"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21885","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50116119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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